Dept of Engineering Maths University of Bristol

In this talk I shall describe recent work inspired by problems in cell biology, namely how the dynamics of small G-proteins underlies polarity formation. Their dynamics is such that their active membrane bound form diffuses more slowly. Hence you might expect Turing patterns. Yet how do cells form backs and fronts or single isolated patches. In understanding these questions we shall show that the key is to identify the parameter region where Turing bifurcations are sub-critical. What emerges is a unified 2-parameter bifurcation diagram containing pinned fronts, localised spots, localised patterns. This diagram appears in many canonical models such as Schnakenberg and Brusselator, as well as biologically more realistic systems. A link is also found between theories of semi-string interaction asymptotics and so-called homoclinic snaking. I will close with some remarks about relevance to root hair formation and to the importance of subcriticality in biology.